The concern that leading Democrats have is that wealthy Americans would move abroad to avoid tax. Hillary was basically opposed to residence-based taxation on this basis.
The left wing of the party, as represented by Bernie Saunders proposed tax relief to Americans overseas, but not eliminating citizenship taxation. The reasoning being not to lose taxes on wealthy Americans living in tax havens.
Unsurprisingly "Democrats Abroad" is strongly in favour of residence-based tax. -source
The relevant section of the Democrat platform states:
Democrats believe that no one should be able avoid[sic] paying their fair share by hiding money abroad, and that corrupt leaders and terrorists should not be able to use the system of international finance to their advantage. We will work to crack down on tax evasion and promote transparency to fight corruption and terrorism. And we will make sure that law-abiding Americans living abroad are not unfairly penalized by finding the right solutions for them to the requirements under the Foreign Account Tax Compliance Act (FATCA) and Report of Foreign Bank and Financial Accounts (FBAR). --source